The fixity of permutation groups
type: Item
instance of: scholarly article
title: The fixity of permutation groups
zbMATH Open document ID: 0832.20002
author: Jan Saxl
author: Aner Shalev
published in: Journal of Algebra
publication date: 1995-08-13T00:00:00Z
review text: By definition, the fixity of a finite permutation group \(G\) is the maximal number of fixed points of a non-identity element of \(G\); so if \(f\) denotes the fixity of \(G\) and \(n\) the degree of \(G\) then \(n-f\) is the minimal degree of \(G\). The authors prove some general theorems on transitive permutation groups \(G\) with given fixity \(f>0\) where \(G\) is supposed in addition to be nilpotent (Theorem 1.1), solvable (Theorem 1.2) or primitive (Theorem 1.3). If \(G\) is nilpotent then \(G\) has a subgroup whose index and nilpotency class are both bounded by a function of \(f\). If \(G\) is primitive then either \(G\) has a solvable subgroup of derived length at most 4 whose index is bounded by a function of \(f\) or \(G\) is almost simple with \(F^*(G)\) similar to \(\text{PSL}(2,q)\) or \(Sz(q)\) in its natural doubly-transitive representation. Of course, the last-mentioned deep theorem depends on the classification of finite simple groups.
Mathematics Subject Classification ID: 20B15
Mathematics Subject Classification ID: 20B05
Mathematics Subject Classification ID: 20B20
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D05
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 783908
zbMATH Keywords: finite permutation groups
zbMATH Keywords: nilpotent groups
zbMATH Keywords: solvable groups
zbMATH Keywords: primitive permutation groups
zbMATH Keywords: fixity
zbMATH Keywords: numbers of fixed points
zbMATH Keywords: minimal degree
zbMATH Keywords: transitive permutation groups
zbMATH Keywords: derived length
zbMATH Keywords: classification of finite simple groups
MaRDI profile type: Publication
full work available at URL: jabr.1995.1171
OpenAlex ID: W2008685341
DOI: 10.1006/JABR.1995.1171
description: scientific article; zbMATH DE number 783908